Need to find the Ricci scalar curvature of this metric

[chinared]

Homework Statement

Need to find the Ricci scalar curvature of this metric:
ds² = e^2a(z)(dx² + dy²) + dz² − e^2b(z)dt²

Homework Equations

The Attempt at a Solution

I tried to find the solution, but failed to pass the calculation of Riemann curvature tensor:

<The Christoffel connection> Here a'(z) denotes the first derivative of a(z) respect to z.
$\Gamma\stackrel{x}{xz}$=$\Gamma\stackrel{x}{zx}$=a'(z)
$\Gamma\stackrel{y}{yz}$=$\Gamma\stackrel{y}{zy}$=a'(z)
$\Gamma\stackrel{z}{tt}$=b'(z)e^2b(z)
$\Gamma\stackrel{z}{xx}$=$\Gamma\stackrel{z}{yy}$=-a'(z)e^2a(z)
$\Gamma\stackrel{t}{tz}$=$\Gamma\stackrel{t}{zt}$=b'(z)
$\Gamma\stackrel{}{either}$=0

<The Riemann curvature tensor>
$\R\stackrel{x}{zxz}$=$\R\stackrel{y}{zyz}$=-a''(z)-[a'(z)]²
$\R\stackrel{z}{tzt}$=b''(z)+[b'(z)]²

I tried to find the Ricci scalar curvature(R) from current result, but it gave a function depend on z. Is there any problem in my calculation?

Thanks for answering this question~!

 

[clamtrox]

I tried to find the Ricci scalar curvature(R) from current result, but it gave a function depend on z. Is there any problem in my calculation?

Why wouldn't it depend on z when your metric does. You can compare this to the FRW-metric. In that case, the metric and the scalar curvature depend on time.

The results look reasonable but I'm a bit too lazy to check it explicitly.